Factor the following expression: $-3$ $x^2$ $-1$ $x+$ $14$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-3)}{(14)} &=& -42 \\ {a} + {b} &=& & & {-1} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-42$ and add them together. Remember, since $-42$ is negative, one of the factors must be negative. The factors that add up to ${-1}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-7}$ and ${b}$ is ${6}$ $ \begin{eqnarray} {ab} &=& ({-7})({6}) &=& -42 \\ {a} + {b} &=& {-7} + {6} &=& -1 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-3}x^2 {-7}x +{6}x +{14} $ Group the terms so that there is a common factor in each group: $ ({-3}x^2 {-7}x) + ({6}x +{14}) $ Factor out the common factors: $ x(-3x - 7) - 2(-3x - 7) $ Notice how $(-3x - 7)$ has become a common factor. Factor this out to find the answer. $(-3x - 7)(x - 2)$